Context. Radio observations have revealed magnetic fields in the intracluster medium (ICM) of galaxy clusters, and their energy density is nearly in equipartition with the turbulent kinetic energy. This suggests magnetic field amplification by dynamo processes during cluster formation. However, observations are limited to redshifts z ≲ 0.7, and the weakly collisional nature of the ICM complicates studying magnetic field evolution at higher redshifts through theoretical models and simulations. Aims. Using a model of the weakly collisional dynamo, we modelled the evolution of the Faraday rotation measure (RM) in galaxy clusters of different masses, up to z ≃ 1.5, and investigated its properties such as its radial distribution up to the virial radius r200. We compared our results with radio observations of various galaxy clusters. Methods. We used merger trees generated by the modified GALFORM algorithm to track the evolution of plasma quantities during galaxy cluster formation. Assuming the magnetic field remains in equipartition with the turbulent velocity field, we generated RM maps to study their properties. Results. We find that both the standard deviation of RM, σRM, and the absolute average |μRM| increase with cluster mass. Due to redshift dilution, RM values for a fixed cluster mass remain nearly constant between z = 0 and z = 1.5. For r/r200 ≳ 0.4, σRM does not vary significantly with L/r200, with L being the size of the observed RM patch. Below this limit, σRM increases as L decreases. We find that radial RM profiles have a consistent shape, proportional to 10-1.2(r/r200), and are nearly independent of redshift. Our z = 0 profiles for Mclust = 1015 M⊙ match RM observations in the Coma cluster but show discrepancies with Perseus, possibly due to high gas mixing. Models for clusters with Mclust = 1013 and 1015 M⊙ at z = 0 and z = 0.174 align well with Fornax and A2345 data for r/r200 ≲ 0.4. Our model can be useful for generating mock polarization observations for current and next-generation radio telescopes.